Problem: Simplify and expand the following expression: $ \dfrac{5z}{z - 5}+\dfrac{z + 4}{z + 2} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(z - 5)(z + 2)$ Multiply the first term by $\dfrac{z + 2}{z + 2}$ $ \begin{align*} \dfrac{5z}{z - 5} \times \dfrac{z + 2}{z + 2} & = \dfrac{(5z)(z + 2)}{(z - 5)(z + 2)} \\ & = \dfrac{5z^2 + 10z}{(z - 5)(z + 2)}\end{align*} $ Multiply the second term by $\dfrac{z - 5}{z - 5}$ $ \begin{align*} \dfrac{z + 4}{z + 2} \times \dfrac{z - 5}{z - 5} & = \dfrac{(z + 4)(z - 5)}{(z + 2)(z - 5)} \\ & = \dfrac{z^2 - z - 20}{(z + 2)(z - 5)}\end{align*} $ Now we have: $ = \dfrac{5z^2 + 10z}{(z - 5)(z + 2)} + \dfrac{z^2 - z - 20}{(z + 2)(z - 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5z^2 + 10z + z^2 - z - 20}{(z - 5)(z + 2)} $ $ = \dfrac{6z^2 + 9z - 20}{(z - 5)(z + 2)}$ Expand the denominator: $ = \dfrac{6z^2 + 9z - 20}{z^2 - 3z - 10}$